Read More
Date: 30-3-2019
4818
Date: 19-5-2019
2371
Date: 16-5-2018
1226
|
Kummer's first formula is
(1) |
where is the hypergeometric function with , , , ..., and is the gamma function. The identity can be written in the more symmetrical form as
(2) |
where and is a positive integer (Bailey 1935, p. 35; Petkovšek et al. 1996; Koepf 1998, p. 32; Hardy 1999, p. 106). If is a negative integer, the identity takes the form
(3) |
(Petkovšek et al. 1996).
Kummer's second formula is
(4) |
|||
(5) |
|||
(6) |
where is a Whittaker function, is the confluent hypergeometric function of the first kind, is a Pochhammer symbol, is a modified Bessel function of the first kind, and , , , ....
REFERENCES:
Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.
Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.
Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 42-43 and 126, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.
|
|
دور النظارات المطلية في حماية العين
|
|
|
|
|
العلماء يفسرون أخيرا السبب وراء ارتفاع جبل إيفرست القياسي
|
|
|
|
|
اختتام المراسم التأبينية التي أهدي ثوابها إلى أرواح شهداء المق*ا*و*مة
|
|
|